Bose-Einstein condensation in diamond hierarchical lattices

The Bose-Einstein condensation of non-interacting particles restricted to move on the sites of hierarchical diamond lattices is investigated. Using a tight-binding single particle Hamiltonian with properly rescaled hopping amplitudes, we are able to employ an orthogonal basis transformation to exactly map it on a set of decoupled linear chains with sizes and degeneracies written in terms of the network branching parameter $q$ and generation number $n$. The integrated density of states is shown to have a fractal structure of gaps and degeneracies with a power-law decay at the band bottom. The spectral dimension $d_s$ coincides with the network topological dimension $d_f=\ln{(2q)}/\ln{(2)}$. We perform a finite-size scaling analysis of the fraction of condensed particles and specific heat to characterize the critical behavior of the BEC transition that occurs for $q>2$ ($d_s>2$). The critical exponents are shown follow those for lattices with a pure power-law spectral density, with non-mean-field values for $q<8$ ($d_s<4$). The transition temperature is shown to grow monotonically with the branching parameter, obeying the relation $1/T_c = a + b/(q-2)$.